Central Limit Theorem
- Animation
- R code

Central Limit Theorem

Let $\normalsize X_{1}, X_{2}, \ldots$ be a sequence of iid random variables with $\normalsize E(X_{i})=\mu$ and $\normalsize 0<Var(X_i)=\sigma^2<\infty$ . Define $\normalsize \bar{X}_n=(1/n)\sum_{i=1}^n X_i$ . Let $\normalsize G_n(x)$ denote the cdf of $\normalsize \sqrt{n}(\bar{X}_n-\mu)/\sigma$ . Then, for any $\normalsize x$ , $\normalsize -\infty<x<\infty$ ,

$\lim_{n\rightarrow\infty}G_n(x)=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}}e^{-y^2/2}dy$

that is, $\normalsize \sqrt{n}(\bar{X}_n-\mu)/\sigma$ has a limiting standard normal distribution.

Animation

Note that from $\normalsize n \geq 15$ , the P-value becomes large, which indicates that we cannot reject the null hypothesis (the normality of the sample mean). The population distribution of the random variables is exponential distribution $\normalsize \mathbf{Exp}(1)$ . For small $\normalsize n$ , we can still observe the characteristic of the exponential distribution: the density is right-skewed. But as $\normalsize n$ grows large, the distribution tends to be symmetrical!

Demonstration of the Central Limit Theorem

loading animation frames... 0%

View frame /100 Loop Time Interval: Step:

This animation shows the distribution of the sample mean as the sample size $\normalsize n$ grows. A normality test shapiro.test() is also performed to check whether the sample mean follows the normal distribution.

R code

oopt = ani.options(ani.height = 500, ani.width = 600, outdir = getwd(), nmax = 100,
    interval = 0.1, title = "Demonstration of the Central Limit Theorem",
    description = "This animation shows the distribution of the sample
    mean as the sample size grows.")
ani.start()
par(mar = c(3, 3, 1, 0.5), mgp = c(1.5, 0.5, 0), tcl = -0.3)
clt.ani(type = "h")
ani.stop()
ani.options(oopt)